Example 1

Example 2

Example 3

Example 4

First, we write out the prime factorization of each denominator. That's
going to give us 3 x 7 and 2 x 3 x 5. To get our LCD, we simply multiply one of each type of prime number together.

So we'll need to hang onto our 3 x 7, but we can eliminate the 3 from the second set of prime factors. Our LCD will be 3 x 7 x 2 x 5 = 210. Not that small of a number, but it is smaller than it would be if we simply took 21 x 30.

Example 5

Find the LCD of and .

First write out the prime factorizations of the denominators:

and

As
before, we've got exactly one repeat 3, so we can throw one of them
out. Therefore, rather than taking 2 x 2 x 3 x 3 x 3 x 5, we just have
to take 2 x 2 x 3 x 3 x 5. Only slightly less work, but we'll take it.
It's not like we're making commission on this.

LCD = 2 x 2 x 3 x 3 x 5 = 180

Example 6

Find the LCD of and .

Write out the prime factorizations of the denominators:

and .

In
this case, the denominators have no factors in common. (Which makes it a
pretty sure bet they're going to hook up.) In this case, the LCD is
merely the product of the two denominators: 510. That's a pretty big
denominator, but we'll have to live with it. We should at least ask it
to start paying rent.

Example 7

Rewrite and as fractions whose denominator is the LCD of and .

You can figure out the LCD for these fractions in one of two ways.
One: you can do the legwork, or two: you can look back up the page and
realize that we already tried this one. Keen observation will get you
far in this world.

We already know that the LCD for these two
fractions is 180. That means that, instead of and , we will
instead be working with the equivalent fractions of and .
Please show them to their cubicles.